A Parametrization for Triad Interactions of Internal Gravity Waves in Varying Background Flows for WKBJ Ray-Tracing Methods

Internal gravity waves are a well known mechanism of energy transport in stratified fluids such as the atmosphere and the ocean. Their abundance and importance for various geophysical processes like ocean mixing and momentum deposition in atmospheric jets are widely accepted. While resonant wave-wave interactions of monochromatic disturbances have received intensive study, little work has been done on triad interactions between wave trains that are modulated by a variable mean flow.

Using the method of multiple scale asymptotics we consider a weakly non-linear Boussinesq WKBJ theory for interacting gravity wave trains propagating through a finite amplitude background flow. Consequently the wave trains are allowed to spectrally pass through resonance conditions and exchange energy when sufficiently close to resonance. We find a global optimal threshold for the deviation from resonance and derive a corresponding parametrization for the triad interaction applicable to ray tracing schemes.

We test the theory with idealized simulations in which two wave trains generate a third by passing through resonance in a sinusoidal background shear flow with varying vertical scales. Comparing WKBJ simulations with wave resolving large eddy simulations we find qualitative and quantitative agreement. Furthermore we assess the impact of the strength of the modulation as well as the effect of the wave amplitudes on the energy exchange between the interacting wave triad.